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In this paper a new method for inverting the Laplace transform from the real axis is formulated. This method is based on a quadrature formula. We assume that the unknown function $f(t)$ is continuous with (known) compact support. An…
In this paper, we consider an inverse problem to determine a source term in a parabolic equation, where the data are obtained at a certain time. In general, this problem is ill-posed, therefore the Tikhonov regularization method is proposed…
We study the Taylor expansion for the solution of a differential equation driven by a multidimensional Holder path with exponent \beta> 1/2. We derive a convergence criterion that enables us to write the solution as an infinite sum of…
In this research paper, we present an exact matrix form analytical solution of the multi-dimensional generalized Langevin equation with quadratic potentials. Our investigation provides detailed expressions for the two-dimensional…
This paper examines solutions to the Laplace equation using analytical techniques, including separation of variables and the Poisson integral formula, and probabilistic methods, such as Brownian motion. We address applications to imaging,…
In this paper, we obtain the analytical solutions of Laplace transforms based some novel integrals with suitable convergence conditions, by using hypergeometric approach (some algebraic properties of Pochhammer symbol and classical…
One of the main objectives of science is the recognition of a general pattern in a particular phenomenon in some particular regime. In this work, this is achieved with the analytical expression for the optimal protocol that minimizes the…
The Lambert W function has utility for solving various exponential and logarithmic equations arranged in the form of $g(x)e^{g(x)}$. Using the Lambert W function and tetration, a variety of categorized inversion formulas are presented.…
We use the middle convolution to obtain some old and new algebraic solutions of the Painlev\'e VI equations.
A discrete Laplace transform and its inversion formula are obtained by using a quadrature of the continuous Fourier transform which is given in terms of Hermite polynomials and its zeros. This approach yields a convergent discrete formula…
We present a simple method to obtain the solution of a few orbital problems: the Kepler problem, the modified Kepler problem by the addition of an inverse square potential and linear force.
Burgers' equation with fixed Dirichlet boundary conditions is considered on generic bounded intervals. By using the Hopf-Cole transformation and the exact operational solution recently established for linear reaction-diffusion equations…
We consider a coefficient inverse problem for the dielectric permittivity in Maxwell's equations, with data consisting of boundary measurements of one or two backscattered or transmitted waves. The problem is treated using a Lagrangian…
We develop a differential theory for the polarity transform parallel to that for the Legendre transform, which is applicable when the functions studied are "geometric convex", namely convex, non-negative and vanish at the origin. This…
A generalization of the Laplace transform based on the generalized Tsallis $q$-exponential is given in the present work for a new type of kernel. We also define the inverse transform for this generalized transform based on the complex…
The second order $N$-dimensional Schr\"odinger equation with pseudoharmonic potential is reduced to a first order differential equation by using the Laplace transform approach and exact bound state solutions are obtained using convolution…
Addition formulae of trigonometric and elliptic functions are used to generate B\"acklund transformations together with their connecting quadrilateral equations. As a result we obtain periodic solutions for a number of multidimensionally…
We give an algorithm to compute $N$ steps of a convolution quadrature approximation to a continuous temporal convolution using only $O(N \log N)$ multiplications and $O(\log N)$ active memory. The method does not require evaluations of the…
We compute the full set of second-order inertial corrections to the instantaneous force and torque acting on a small spherical rigid particle moving unsteadily in a general steady linear flow. This is achieved by using matched asymptotic…
This paper presents a novel way of computing front positions in Fisher-KPP equations. Our method is based on an exact relation between the Laplace transform of the initial condition and some integral functional of the front position. Using…